552 
ON THREE-BAR MOTION. 
[623 
trinodal sextic; viz. in the Three-Bar Curve the two fixed points are foci, and they 
determine a third focus * ; and the condition is that the nodes are situate on the circle 
through the 3 foci. 
The nodes are two of them arbitrary points on the circle; and the third of them 
is a point such that, measuring the distances along the circle from any fixed point 
of the circumference, the sum of the distances of the nodes is equal to the sum of 
the distances of the foci. Considering the two fixed points as given, the curve 
depends upon five parameters, viz. the lengths of the connecting bars and the sides of 
the triangle. Taking the form of the triangle as given, there are then only three 
parameters, say the lengths of the connecting bars and the base of the triangle; in 
this case the third focus is determined, and therefore the circle through the three 
foci; we may then take two of the nodes as given points on this circle, and thereby 
establish two relations between the three parameters, in fact, we thereby determine 
the differences of the squares of the lengths in question : but the third node is then 
an absolutely determined point on the circle, and we cannot make use of it for com 
pleting the determination of the parameters ; viz. one parameter remains arbitrary. Or, 
what is the same thing, given the three foci and also the three nodes, consistently 
with the foregoing conditions, viz. the nodes lie in the centre through the three foci, 
the sum of the distances of the nodes being equal to the sum of the distances of 
the foci: we have a singly infinite series of three-bar curves. 
In reference to the notation proper for the theorem of the triple generation, I 
shall, when only a single node of generation is attended to, take the curve to be 
generated as shown in the annexed Figure 1; viz. 0 is the generating point, OC l B 1 
the triangle, G, B the fixed points, GG 1 and BB 1 the radial bars. The sides of the 
Fig. l. 
o 
triangle are a x , b 1} c x ; its angles are 0, = A, B l} —B, G lt = G: the bars CC 1 and BB X 
are =a 2 and a 3 respectively, and the distance CB is = a. The sides a lt b x , c x may be 
put = hi (sin A, sin B, sin G), and the lines a lf a. 2 , a s =(k lf k.,, & 3 ) sin A, viz. the original 
data a 1? b ly c 1} a 1} a.,, a 3 , may be replaced by the angles A, B, G (A + B + C = tt) and 
the lines k ly k 2 , k s . And it is convenient to mention at once that the third focus 
A is then a point such that ABC is a triangle similar and congruent to 0B 1 G 1 . 
* A focus is a point, given as the intersection of a tangent to the curve from one circular point at infinity 
with a tangent from the other circular point at infinity; if the circular points are simple or multiple points on 
the curve, then the tangent or tangents at a circular point should be excluded from the tangents from the 
point; and the intersection of two such tangents at the two circular points respectively is not an ordinary 
focus; but, as the points in question are the only kind of foci occurring in the present paper, I have in the 
text called them foci.